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Proof step 17 from proof bytes2int-size-01 (previous step)

2 ^ ((n→i(# bytes1) + 1) * 8 - 1) ≤ bytes2int(i2b(0) ' + by ' + bytes1), b2i(by) < 0, 256 * bytes2int(i2b(0) ' + by ' + bytes1) + band(255, b2i(by0)) < 2 ^ (8 * (1 + n→i(# bytes1)) - 1) * 256, ∀ by, bytes0. # bytes0 < #(bytes1 + by0 ') → true → b2i(by) < 0 → 2 ^ ((n→i(# bytes0) + 1) * 8 - 1) ≤ bytes2int(i2b(0) ' + by ' + bytes0) ⊦

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 18:
j * i + j0 < j * i0, abs(j0) < j ⊦ i ≤ i0
Proof step 19:
2 ^ ((n→i(# bytes1) + 1) * 8 - 1) ≤ bytes2int(i2b(0) ' + by ' + bytes1), b2i(by) < 0, 256 * bytes2int(i2b(0) ' + by ' + bytes1) + band(255, b2i(by0)) < 2 ^ (8 * (1 + n→i(# bytes1)) - 1) * 256, ∀ by, bytes0. # bytes0 < #(bytes1 + by0 ') → true → b2i(by) < 0 → 2 ^ ((n→i(# bytes0) + 1) * 8 - 1) ≤ bytes2int(i2b(0) ' + by ' + bytes0) ⊦ 256 * bytes2int(i2b(0) ' + by ' + bytes1) + band(255, b2i(by0)) < 256 * 2 ^ (8 * (1 + n→i(# bytes1)) - 1) ∧ abs(band(255, b2i(by0))) < 256

Rule argument

seq: j * i + j0 < j * i0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, j0, i0] → [256, bytes2int(i2b(0) ' + by ' + bytes1), band(255, b2i(by0)), 2 ^ (8 * (1 + n→i(# bytes1)) - 1)] rotate: F spec: int-mult inst: name: modlsp-02

Additional information

Used simplifier rules: ⊦ i + j = j + i;% ⊦ i * j = j * i;% ⊦ (bytes + bytes0) + bytes1 = bytes + bytes0 + bytes1;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ (* i2) * i + j < (* i1) * i ↔ j < (i1 - i2) * i;% ⊦ i * 0 = 0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 ≤ i → ¬ band(i, j) < 0;% ⊦ band(i, band(i0, j)) = band(band(i, i0), j);% ⊦ band(i, j) = band(j, i);% ⊦ i + j + i0 = (i + j) + i0; Used terms: 256, bytes2int(i2b(0) ' + by ' + bytes1), band(255, b2i(by0)), 2 ^ (8 * (1 + n→i(# bytes1)) - 1)